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<h1 class="heading"><a href="MATH-2023-OPDE.html"><span class="title">MATH 2023: Ordinary and Partial Differential Equations</span></a></h1>
<p class="byline">Xiaoyi Chen and Wei Zhang</p>
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<li class="link frontmatter"><a href="meta_frontmatter.html" data-scroll="meta_frontmatter" class="internal"><span class="title">Front Matter</span></a></li>
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<a href="ch_first.html" data-scroll="ch_first" class="internal"><span class="codenumber">1</span> <span class="title">Introduction</span></a><ul>
<li><a href="sec_1-intro.html" data-scroll="sec_1-intro" class="internal">Classification of Differential Equations</a></li>
<li><a href="sec_2-intro.html" data-scroll="sec_2-intro" class="internal">Linear and Nonlinear Equation</a></li>
<li><a href="sec_3-intro.html" data-scroll="sec_3-intro" class="internal">Geometrical Aspect</a></li>
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<a href="ch_second.html" data-scroll="ch_second" class="internal"><span class="codenumber">2</span> <span class="title">First Order Ordinary Differential Equations</span></a><ul>
<li><a href="sec2_1.html" data-scroll="sec2_1" class="internal">Linear Equations</a></li>
<li><a href="sec2_2.html" data-scroll="sec2_2" class="internal">Further Discussion of Linear Equations (For reading only)</a></li>
<li><a href="sec2_3.html" data-scroll="sec2_3" class="internal">Separable Equations</a></li>
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<a href="ch_third.html" data-scroll="ch_third" class="internal"><span class="codenumber">3</span> <span class="title">third Order Linear Equations</span></a><ul>
<li><a href="sec3_1.html" data-scroll="sec3_1" class="internal">Homogeneous equations with constant coefficient</a></li>
<li><a href="sec3_2.html" data-scroll="sec3_2" class="internal">Fundamental Solutions of Linear Homogeneous Equations</a></li>
<li><a href="sec3_3.html" data-scroll="sec3_3" class="internal">Linear Independence and Wronskian</a></li>
<li><a href="sec3_4.html" data-scroll="sec3_4" class="internal">Complex roots of the characteristic equations</a></li>
<li><a href="sec3_5.html" data-scroll="sec3_5" class="internal">Repeated Roots: Reduction of Order</a></li>
<li><a href="sec3_6.html" data-scroll="sec3_6" class="internal">Non-homogeneous Equations and Method of Undetermined Coefficients</a></li>
<li><a href="sec3_7.html" data-scroll="sec3_7" class="internal">Variation of Parameters</a></li>
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<a href="ch_four.html" data-scroll="ch_four" class="internal"><span class="codenumber">4</span> <span class="title">Higher Order Linear Equations</span></a><ul>
<li><a href="sec4_1.html" data-scroll="sec4_1" class="internal">General Theory of the <span class="process-math">\(n\)</span>-th Order Linear Equations</a></li>
<li><a href="sec4_2.html" data-scroll="sec4_2" class="internal">Homogeneous Equations with Constant Coefficients</a></li>
<li><a href="sec4_3.html" data-scroll="sec4_3" class="internal">The Method of Undetermined Coefficients</a></li>
<li><a href="sec4_4.html" data-scroll="sec4_4" class="internal">The Method of Variation of Parameters</a></li>
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<a href="ch_five.html" data-scroll="ch_five" class="internal"><span class="codenumber">5</span> <span class="title">Series Solutions of Second Order Linear Equations</span></a><ul>
<li><a href="sec5_1.html" data-scroll="sec5_1" class="internal">Brief Review on Power Series</a></li>
<li><a href="sec5_2.html" data-scroll="sec5_2" class="internal">Introduction</a></li>
<li><a href="sec5_3.html" data-scroll="sec5_3" class="internal">Series Solutions Near an Ordinary Point</a></li>
<li><a href="sec5_4.html" data-scroll="sec5_4" class="internal">Euler’s Equation</a></li>
<li><a href="sec5_5.html" data-scroll="sec5_5" class="internal">Series Solution near a Regular Singular Point</a></li>
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<a href="ch_six.html" data-scroll="ch_six" class="internal"><span class="codenumber">6</span> <span class="title">System of First Order Linear Equations</span></a><ul>
<li><a href="sec6_1.html" data-scroll="sec6_1" class="internal">Introduction <span class="process-math">\(\&amp;\)</span> Basic Theory</a></li>
<li><a href="sec6_2.html" data-scroll="sec6_2" class="internal">Homogeneous System with Constant Coefficients</a></li>
<li><a href="sec6_3.html" data-scroll="sec6_3" class="internal">Complex Eigenvalues</a></li>
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<li><a href="sec6_5.html" data-scroll="sec6_5" class="internal">Fundamental Matrices</a></li>
<li><a href="sec6_6.html" data-scroll="sec6_6" class="internal">Non-homogeneous linear systems</a></li>
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<a href="ch_seven.html" data-scroll="ch_seven" class="internal"><span class="codenumber">7</span> <span class="title">Partial Differential Equations</span></a><ul>
<li><a href="sec7_1.html" data-scroll="sec7_1" class="internal">Two-Point Boundary Value Problems</a></li>
<li><a href="sec7_2.html" data-scroll="sec7_2" class="internal">Eigenvalue Problems</a></li>
<li><a href="sec7_3.html" data-scroll="sec7_3" class="internal">Fourier Series</a></li>
<li><a href="sec7_4.html" data-scroll="sec7_4" class="internal">The Fourier Convergence Theorem</a></li>
<li><a href="sec7_5.html" data-scroll="sec7_5" class="internal">Even and Odd Functions</a></li>
<li><a href="sec7_6.html" data-scroll="sec7_6" class="internal">Introduction to Partial Differential Equations</a></li>
<li><a href="sec7_7.html" data-scroll="sec7_7" class="internal">1D Heat Equation; Solutions by Separation of Variable and Fourier Series</a></li>
<li><a href="sec7_8.html" data-scroll="sec7_8" class="internal">Other Heat Conduction Problems</a></li>
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<a href="ch_eight.html" data-scroll="ch_eight" class="internal"><span class="codenumber">8</span> <span class="title">Laplace transform</span></a><ul>
<li><a href="sec8_1.html" data-scroll="sec8_1" class="internal">What are Laplace Transforms, and Why?</a></li>
<li><a href="sec8_2.html" data-scroll="sec8_2" class="internal">Finding Laplace Transforms</a></li>
<li><a href="sec8_3.html" data-scroll="sec8_3" class="internal">Finding inverse transforms using partial fractions</a></li>
<li><a href="sec8_4.html" data-scroll="sec8_4" class="internal">Solving ODEs and ODE Systems</a></li>
<li><a href="sec8_5.html" data-scroll="sec8_5" class="active">Step input and Impulse problems</a></li>
<li><a href="sec8_6.html" data-scroll="sec8_6" class="internal">Laplace transform for PDE (heat equation)</a></li>
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<a href="ch_features.html" data-scroll="ch_features" class="internal"><span class="codenumber">9</span> <span class="title">Examples of PreTeXt features</span></a><ul><li><a href="sec_features-blocks.html" data-scroll="sec_features-blocks" class="internal">Environments and Blocks</a></li></ul>
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<li class="link"><a href="solutions-1.html" data-scroll="solutions-1" class="internal"><span class="codenumber">A</span> <span class="title">Selected Hints</span></a></li>
<li class="link"><a href="solutions-2.html" data-scroll="solutions-2" class="internal"><span class="codenumber">B</span> <span class="title">Selected Solutions</span></a></li>
<li class="link"><a href="appendix-1.html" data-scroll="appendix-1" class="internal"><span class="codenumber">C</span> <span class="title">List of Symbols</span></a></li>
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<main class="main"><div id="content" class="pretext-content"><section class="section" id="sec8_5"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">8.5</span> <span class="title">Step input and Impulse problems</span>
</h2>
<p id="p-480">Laplace transform methods are particularly valuable in handling differential equations involving impulse and step functions. In the following, we first give the definitions for step function and Delta function and then give examples involving these two types of functions.</p>
<section class="subsubsection" id="step-function-and-delta-function"><h3 class="heading hide-type">
<span class="type">Subsubsection</span> <span class="codenumber">8.5.1</span> <span class="title">Step function and Delta function</span>
</h3>
<p id="p-481">The unit step function is defined by</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u_c(t)=\begin{cases}
0,\quad t&lt;c,\\
1,\quad t \geq c.
\end{cases}
\end{equation*}
</div>
<p class="continuation">The graph of <span class="process-math">\(y=u_c(t)\)</span> is given in Figure <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "Fig1_1" missing or not unique]</code>.</p>
<p id="p-482"></p>
<p id="p-483">The Laplace transform of <span class="process-math">\(u_c(t)\)</span> is easily determined</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}[u_c(t)]=\int_0^{\infty} e^{-st} u_c(t) \mathrm{d}t=\int_c^{\infty} e^{-st} \mathrm{d}t=\frac{e^{-cs}}{s},\quad s&gt;0.
\end{equation*}
</div>
<p id="p-484">For the Delta function, we may understand it in the sense of limit. Define a sequence of functions</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
d_{\tau}(t)=\begin{cases}
1/(2\tau),\quad - \tau&lt;t&lt;\tau,\\
0,\quad t \leq -\tau ~\mathrm{or} ~t \geq \tau,
\end{cases}
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(\tau\)</span> is a small positive constant (see the illustration of <span class="process-math">\(d_{\tau}(t)\)</span> in Figure <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "Fig2_1" missing or not unique]</code>).</p>
<p id="p-485"></p>
<p id="p-486">One can see that</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq6_1.html ./knowl/Eq7_1.html" id="Eq6_1">
\begin{equation}
\lim_{\tau \to 0} d_{\tau}(t)=0,\quad t\neq 0.\tag{8.5.1}
\end{equation}
</div>
<p class="continuation">And let</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq6_1.html ./knowl/Eq7_1.html">
\begin{equation*}
I(\tau)=\int_{-\infty}^{\infty} d_{\tau}(t) \mathrm{d}t,
\end{equation*}
</div>
<p class="continuation">then it is to calculate <span class="process-math">\(I(\tau)=1\)</span> for each <span class="process-math">\(\tau \neq 0\)</span> and</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq6_1.html ./knowl/Eq7_1.html" id="Eq7_1">
\begin{equation}
\lim_{\tau \to 0} I(\tau)=1.\tag{8.5.2}
\end{equation}
</div>
<p class="continuation">Equations (<a href="" class="xref" data-knowl="./knowl/Eq6_1.html" title="Equation 8.5.1">(8.5.1)</a>) and (<a href="" class="xref" data-knowl="./knowl/Eq7_1.html" title="Equation 8.5.2">(8.5.2)</a>) can be used to define the unit impulse function <span class="process-math">\(\delta\)</span> which have the properties</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq6_1.html ./knowl/Eq7_1.html">
\begin{equation*}
\delta(t)=0,\quad t \neq 0
\end{equation*}
</div>
<p class="continuation">and</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq6_1.html ./knowl/Eq7_1.html">
\begin{equation*}
\int_{-\infty}^{\infty} \delta(t) \mathrm{d} t=1.
\end{equation*}
</div></section><section class="subsubsection" id="step-input-problems"><h3 class="heading hide-type">
<span class="type">Subsubsection</span> <span class="codenumber">8.5.2</span> <span class="title">Step Input problems</span>
</h3>
<p id="p-487">Here we consider the following initial value problem which involves a step input function—typical of many control-type problems:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\ddot{x}+3\dot{x}+2x=u_0(t), \ \  x(0)=\dot{x}(0) =0. \label{Hsys}
\end{equation*}
</div>
<p class="continuation">Here we have a second order ODE representing a system that is at rest until time <span class="process-math">\(t=0,\)</span> when a unit step input <span class="process-math">\(u_0(t)\)</span> is applied; we seek the output <span class="process-math">\(x(t).\)</span></p>
<p id="p-488"><dfn class="terminology">Example</dfn></p>
<ol id="p-489" class="decimal">
<li id="li-84">
<p id="p-490">Transform (<code class="code-inline tex2jax_ignore">[cross-reference to target(s) "Hsys" missing or not unique]</code>) to get</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
(s^2+3s+2)X(s) = \frac{1}{s},
\end{equation*}
</div>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
X(s) = \frac{1}{s}\cdot\frac{1}{(s+2)(s+1)}=
\frac{1}{s}\left(\frac{-1}{s+2}+\frac{1}{s+1}\right).
\end{equation*}
</div>
</li>
<li id="li-85">
<p id="p-491">Apply the Integration property to obtain</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
x(t)&amp;=&amp;\int_0^t \left(  -e^{-2\theta}+e^{-\theta} \right) \, d\theta\\
&amp;=&amp; \left[\frac{1}{2}e^{-2\theta}-e^{-\theta} \right]_0^t\\
&amp;=&amp;\frac{1}{2}e^{-2t}-e^{-t} -\frac{1}{2}+1\\
&amp;=&amp;\frac{1}{2}e^{-2t}-e^{-t} +\frac{1}{2}.
\end{aligned}
\end{equation*}
</div>
</li>
<li id="li-86"><p id="p-492">Show that the solution <span class="process-math">\(x(t)\)</span> is asymptotic to <span class="process-math">\(x=\frac{1}{2},\)</span> why is this obvious as a steady state solution?</p></li>
</ol></section><section class="subsubsection" id="impulse-problem"><h3 class="heading hide-type">
<span class="type">Subsubsection</span> <span class="codenumber">8.5.3</span> <span class="title">Impulse problem</span>
</h3>
<p id="p-493">The problem in the examples below represents the dynamics of a point, initially at rest, moving away from the origin along the <span class="process-math">\(y\)</span>-axis under a constant acceleration of value <span class="process-math">\(10\)</span> for <span class="process-math">\(0\leq t
&lt;1\)</span> and an extra impulse acceleration of size <span class="process-math">\(10\)</span> is applied at <span class="process-math">\(t=1.\)</span> This is like a simple rocket boost, but can you solve it any other way? We use the Dirac impulse function <span class="process-math">\(\delta(t-a)\)</span> which is nonzero at <span class="process-math">\(t=a,\)</span> but zero elsewhere while having unit total area under it:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\delta(t-a)=0 \ {\rm if} \ (t\neq a) \ \
{\rm and} \ \int_{-\infty}^{\infty} \delta(t-a) \, dt = 1.
\end{equation*}
</div>
<p id="p-494"><dfn class="terminology">Example</dfn></p>
<p id="p-495">Consider the ODE initial value problem given by</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y''=10 + 10 \delta(t-1), \ \ \ y(0)=y'(0)=0.
\end{equation*}
</div>
<ol id="p-496" class="decimal">
<li id="li-87"><p id="p-497">Begin by sketching the graph of the acceleration, <span class="process-math">\(y'',\)</span> to show the step increase.</p></li>
<li id="li-88">
<p id="p-498">Transforming according to the table, to get</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
s^2Y -s y(0) -y'(0) = \frac{10}{s}
+ 10 e^{-s}
\end{equation*}
</div>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\rm so, \ rearranging} \ \ \
Y(s)=\frac{10}{s^3} + \frac{10e^{-s}}{s^2} =
5\frac{2}{s^3}+10 e^{-s}\frac{1}{s^2}
\end{equation*}
</div>
</li>
<li id="li-89">
<p id="p-499">From the table use the Delay property to deduce that</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y(t)=5t^2+10(t-1)\,u(t-1)
\end{equation*}
</div>
</li>
<li id="li-90">
<p id="p-500">By interpreting the step function <span class="process-math">\(u(t-1)\)</span> up to and after <span class="process-math">\(t=1,\)</span> show that the impulse at <span class="process-math">\(t=1\)</span> produces what you would expect: a discontinuity in <dfn class="terminology">velocity</dfn> at <span class="process-math">\(t=1.\)</span> Sketch the full solution:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y(t)
=5t^2 \ \ {\rm for} \ t\leq 1 \ \ \ \ \ ({\rm so \ here} \ y'(t)=10t)
\end{equation*}
</div>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y(t) =5t^2 + 10(t-1)\ \ {\rm for} \ t&gt; 1 \ \ \ \
({\rm  so \ here} \ y'(t) =10t +10) .
\end{equation*}
</div>
</li>
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